A sparse hp-finite element method for piecewise-smooth differential equations with periodic boundary conditions
File(s)
Published online version
Author(s)
VandenHeuvel, Daniel
Olver, Sheehan
Type
Journal Article
Abstract
We develop an efficient hp-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and achieve optimal complexity regardless of h or p, both for building the discretisation and solving the resulting linear system in the case where the operator is symmetric positive definite. The basis serves as a useful alternative to other bases such as the Fourier or integrated Legendre bases, especially for problems with discontinuities. We relate the convergence properties of these bases to regions of analyticity in the complex plane, and further use several differential equation examples to demonstrate these properties. The basis spans the low order eigenfunctions of constant coefficient differential operators, thereby achieving better smoothness properties for time-evolution partial differential equations.
Date Issued
2026-02-09
Date Acceptance
2026-01-05
Citation
Numerical Algorithms, 2026
ISSN
1017-1398
Publisher
Springer
Journal / Book Title
Numerical Algorithms
Copyright Statement
© The Author(s) 2026. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
10.1007/s11075-026-02313-y
Publication Status
Published online
Date Publish Online
2026-02-09